As you can see, as v approaches c, v2/c2 approaches 1, and the mass of the object approaches infinity. Clearly an object with infinite mass would be trouble, thus it is speculated that objects with a non-zero rest mass cannot travel faster than light. (Not to mention that it would actually have an undefined mass rather than infinite...)

An interesting (for sci-fi at least) theoretical type of particle is the tachyon which would have a negative rest mass, and consequently it can't move slower than light. I don't believe there has ever been evidence that these actually exist.

Ok, that's right, the mass squared of a tachyon would be negative. Assuming you can't actually have a complex mass value, m0 would have to be complex to cancel the complex value in the denominator. Thus, tachyons have to go faster than light, and slow down as you add energy.

In classic physics, when you throw a rock at 5 mph out of a car moving at 55 mph, the rock is moving at 60 mph to a stationary observer, while it moves at 5 mph to the observer in the car. By this transformation (known as the Galilean transformation for velocities), it would appear that (A) and (B) are moving at 1.5 c with respect to each other - which is greater than the speed of light.

The problem here is that the Galilean transformation is no longer valid and instead the Lorentz transformation is necessary. For one dimension with respect to A:

v' = VA - VB
--------
1 - VAVB
----
C2

v' is the speed which A perceives B is moving at.
Vx is the velocity from the external frame of referencec is the speed of light.
thus:

v' = .75c - -.75c
------------
1 - -.56c2
------
c2

which is: 0.96c.

0.96c is the velocity that A will perceive B is moving at - not 1.5c.

At low speeds, this reduces to the Galilean transformation that we
are quite familiar with and is in line with common sense.

where γ =
1/sqrt(1-v2/c2). Trying to fit this to Newton's
expression for it gives the result that rabidcow quotes. Although
this works for 1-D motion, if you accelerate the particle in any other
direction but parallel to v, you need to change the mass in a
different way (for a example a factor of γ3 is needed
if you accelerate perpendicular to the motion). The variable mass
concept is a helpful idea at first, but only works in very limited
conditions, and will get you into trouble if you try to apply it
further.

Hence the reason an object cannot
be accelerated beyond c should be explained in terms of
momentum, not mass. Newton's Second Law of Motion, as he
originally phrased it in terms of momentum, still holds, providing the
above definition of momentum is used, and requires an infinite force
to accelerate beyond c.

Alternatively, the explanation can be done in terms of the infinite
amount of energy that must be given particles to get them to reach
c (assuming they have mass). You have to use the general form of Einstein's equation:

E = γ m c2

which works for any speed, not just the rest mass energy. Again, we could have done the m = γ m0 trick, but it's unclear which 'mass' is being referred to, and is best avoided for the reason I put above.

Going into the mathematics a bit further, in order to move something you must apply force(F). If you have taken a physics course you probably remember that

F = m*a

or force used is equal to mass accelerated times the acceleration experienced. And this is true, up to a point. What force actually is defined as, however, is the change in momentum with respect to time. In calculus this is expressed as

momentum = m*v

F = d(m*v)/dt

In classical mechanics this equation is equivalent to F=m*a, because the derivative of velocity(v) with respect to time is acceleration(a), and mass(m) is a constant with respect to time. However, because of relativity effects, mass is dependent on velocity according to the equation

m = m0 / sqrt(1 - v^2/c^2)

where m0 is the rest mass of the object, v is the current velocity, and c is the velocity of light in a vacuum. From this it can be seen that as velocity increases the force needed to accelerate the particle also increases. Furthermore, as the velocity approaches c the force required approaches infinity. This means that in order to accelerate an object to lightspeed you would require infinite force; because an infinite force is impossible, it is impossible for matter to travel at the speed of light.

Although matter does not quite move at the constant described as "speed of light" it is still very possible for matter to move at a speed that light also moves at.
The easiest way to get matter to move at the same speed of light or even faster is to simply make light slower. After all, the constant "speed of light" is actually "speed of light in a vacuum" and not "speed of light on earth".

Researchers at the Rowland Institute for Science slowed light to 38 miles per hour in 1999, and other great results have been gotten with the slowing of light since then.
I personally don't do it very often without the help of a train or car, but matter has and does travel quite comfortably at 38 miles per hour (and somewhat faster, taking into account planets moving and all that).

So, depending on your definition on the term "speed of light" matter can indeed not reach that speed, or it quite easily can and does.